It is known that simply connected nonabelian nilpotent Lie groups and not virtually abelian groups of polynomial growth fail to embed bilipschitzly into superreflexive Banach spaces.
We quantify this fact in two ways. First, we provide a lower bound on the distortion of balls in the aforementioned groups into superreflexive spaces. In particular, we show that the $$extract_itex$$L^p$$/extract_itex$$-distortion, $$extract_itex$$(1<p<\infty)$$/extract_itex$$, of a ball of radius $$extract_itex$$n\ge 2$$/extract_itex$$ in the aforementioned groups is exactly $$extract_itex$$(\log n)^{1/\max\{p,2\}}$$/extract_itex$$ up to constants. Second, we characterize the asymptotic behavior of the Lipschitz compression rate of functions from the aforementioned groups into the $$extract_itex$$L^p$$/extract_itex$$ spaces, $$extract_itex$$p>1$$/extract_itex$$. If time permits, I will discuss conjectures on the distortion and compression rate when the target space is $$extract_itex$$L^1$$/extract_itex$$.